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Single Idea 18809
[filed under theme 5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
]
Full Idea
Gentzen's way of formalising logic has accustomed people to the idea that logical truths are simply the by-products of logical rules, that arise when all the assumptions on which a conclusion rests have been discharged.
Clarification
Gentzen's way was 'natural deduction'
Gist of Idea
Logical truths are just the assumption-free by-products of logical rules
Source
Ian Rumfitt (The Boundary Stones of Thought [2015], 2.5)
Book Ref
Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.56
A Reaction
This is the key belief of those who favour the natural deduction account of logic. If you really believe in separate logic truths, then you can use them as axioms.
The
56 ideas
from Ian Rumfitt
11210
|
Standardly 'and' and 'but' are held to have the same sense by having the same truth table
[Rumfitt]
|
11211
|
If a sound conclusion comes from two errors that cancel out, the path of the argument must matter
[Rumfitt]
|
11212
|
The sense of a connective comes from primitively obvious rules of inference
[Rumfitt]
|
11214
|
We learn 'not' along with affirmation, by learning to either affirm or deny a sentence
[Rumfitt]
|
18800
|
Introduction rules give deduction conditions, and Elimination says what can be deduced
[Rumfitt]
|
18802
|
In specifying a logical constant, use of that constant is quite unavoidable
[Rumfitt]
|
18798
|
It is the second-order part of intuitionistic logic which actually negates some classical theorems
[Rumfitt]
|
18799
|
Intuitionists can accept Double Negation Elimination for decidable propositions
[Rumfitt]
|
18803
|
Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables
[Rumfitt]
|
18805
|
Classical logic rules cannot be proved, but various lines of attack can be repelled
[Rumfitt]
|
18804
|
The case for classical logic rests on its rules, much more than on the Principle of Bivalence
[Rumfitt]
|
18807
|
Monotonicity means there is a guarantee, rather than mere inductive support
[Rumfitt]
|
18808
|
Normal deduction presupposes the Cut Law
[Rumfitt]
|
18809
|
Logical truths are just the assumption-free by-products of logical rules
[Rumfitt]
|
18815
|
Logic is higher-order laws which can expand the range of any sort of deduction
[Rumfitt]
|
18813
|
Logical consequence is a relation that can extended into further statements
[Rumfitt]
|
18814
|
'Absolute necessity' would have to rest on S5
[Rumfitt]
|
18816
|
Metaphysical modalities respect the actual identities of things
[Rumfitt]
|
18817
|
We understand conditionals, but disagree over their truth-conditions
[Rumfitt]
|
18819
|
The idea that there are unrecognised truths is basic to our concept of truth
[Rumfitt]
|
18820
|
In English 'evidence' is a mass term, qualified by 'little' and 'more'
[Rumfitt]
|
18821
|
Possibilities are like possible worlds, but not fully determinate or complete
[Rumfitt]
|
18824
|
Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right
[Rumfitt]
|
18825
|
S5 is the logic of logical necessity
[Rumfitt]
|
18826
|
'True at a possibility' means necessarily true if what is said had obtained
[Rumfitt]
|
18827
|
If truth-tables specify the connectives, classical logic must rely on Bivalence
[Rumfitt]
|
18828
|
If two possibilities can't share a determiner, they are incompatible
[Rumfitt]
|
18829
|
The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A
[Rumfitt]
|
18831
|
Medieval logicians said understanding A also involved understanding not-A
[Rumfitt]
|
18830
|
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
[Rumfitt]
|
18834
|
Infinitesimals do not stand in a determinate order relation to zero
[Rumfitt]
|
18835
|
Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics
[Rumfitt]
|
18836
|
A set may well not consist of its members; the empty set, for example, is a problem
[Rumfitt]
|
18837
|
A set can be determinate, because of its concept, and still have vague membership
[Rumfitt]
|
18839
|
An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases
[Rumfitt]
|
18838
|
The extension of a colour is decided by a concept's place in a network of contraries
[Rumfitt]
|
18840
|
When faced with vague statements, Bivalence is not a compelling principle
[Rumfitt]
|
18842
|
Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set
[Rumfitt]
|
18843
|
The iterated conception of set requires continual increase in axiom strength
[Rumfitt]
|
18845
|
If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
[Rumfitt]
|
18846
|
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
|
9390
|
Logic guides thinking, but it isn't a substitute for it
[Rumfitt]
|
9389
|
Vague membership of sets is possible if the set is defined by its concept, not its members
[Rumfitt]
|
17461
|
Some 'how many?' answers are not predications of a concept, like 'how many gallons?'
[Rumfitt]
|
17462
|
A single object must not be counted twice, which needs knowledge of distinctness (negative identity)
[Rumfitt]
|
14532
|
A distinctive type of necessity is found in logical consequence
[Rumfitt, by Hale/Hoffmann,A]
|
12193
|
Logical necessity is when 'necessarily A' implies 'not-A is contradictory'
[Rumfitt]
|
12194
|
Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B'
[Rumfitt]
|
12195
|
Soundness in argument varies with context, and may be achieved very informally indeed
[Rumfitt]
|
12198
|
Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths)
[Rumfitt]
|
12199
|
There is a modal element in consequence, in assessing reasoning from suppositions
[Rumfitt]
|
12201
|
We reject deductions by bad consequence, so logical consequence can't be deduction
[Rumfitt]
|
12202
|
Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not
[Rumfitt]
|
12200
|
A logically necessary statement need not be a priori, as it could be unknowable
[Rumfitt]
|
12203
|
If a world is a fully determinate way things could have been, can anyone consider such a thing?
[Rumfitt]
|
12204
|
The logic of metaphysical necessity is S5
[Rumfitt]
|